I have been given a fundemanetal matrix $e^{At}$ and am asked to find $A$. It's kind of an odd question as we've mainly been calculating $e^{At}$ when given a certain matrix $A$. My question is how do I find $A$ and if you can always find $A$ for any fundamental matrix. Anyways this was given:

\begin{align} e^{At}&= \begin{bmatrix} 2e^{2t}-e^t & e^{2t}-e^t&e^t-e^{2t} \\ e^{2t}-e^t & 2e^{2t}-e^t&e^t-e^{2t} \\ 3e^{2t}-3e^t & 3e^{2t}-3e^t&3e^t-2^{2t} \\ \end{bmatrix} \end{align}


Hint: Take the derivative towards $t$ and plug in $t=0$.

  • $\begingroup$ Yes, that seems to work! Thanks. I'm assuming since $e^{At}=X(t)X(0)^{-1}$ for a fundamental solution $X(t)$ we can always find $A$. $\endgroup$ – J Dijkstra Jan 15 '18 at 0:30
  • $\begingroup$ \begin{align} A&= \begin{bmatrix} 3 & 1&-1 \\ 1 & 3&-1 \\ 3& 3&-1 \\ \end{bmatrix} \end{align} $\endgroup$ – Unknown x Jan 15 '18 at 1:46
  • $\begingroup$ can you please tell me? Am I calculating the correct matrix? $\endgroup$ – Unknown x Jan 15 '18 at 14:07
  • $\begingroup$ @ManeeshNarayanan Correct. (You can also take a look in the answers of odd numbered exercises.) $\endgroup$ – The Phenotype Jan 15 '18 at 17:10
  • $\begingroup$ which textbook? $\endgroup$ – Unknown x Jan 15 '18 at 17:11

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